The present invention relates generally to surface forces measurement instrumentation, and more particularly is a cantilever spring assembly used in probe microscopy.
Scanning Probe Microscopes (SPM""s) are research instruments that have been in use in universities and industrial research laboratories since the early 1980""s. These instruments allow for various imaging of surfaces as well as measurement of the intermolecular forces between two surfaces (or a small tip and a flat surface) in vapors or liquids. The distance resolution is 1 xc3x85, which means that images and forces can be obtained at the atomic level. Over the years, the technique has been improved and its scope extended so that it is now capable of measuring many different surface properties and phenomena.
SPM""s such as AFM""s (Atomic Force Microscope) and STM""s (Scanning Tunneling Microscope) generally consist of a sample surface and a fine round xe2x80x9ctipxe2x80x9d that is supported at the end of a force-measuring cantilever spring. They operate by first bringing (positioning) the tip near the surface and then moving the tip or surface vertically (contact or tapping mode) or laterally (scanning mode) while measuring the force produced on the tip by the surface. The force is calculated by measuring the deflection of the cantilever spring supporting the tip. These displacements are measured by one of three methods: (1) The most common method is the optical or beam deflection method (bouncing a laser light beam from the end of the cantilever spring and measuring its deflection from where it falls on a quadrant photo detector). (2) A less common method is the resistive method (using resistance, semi-conductor or piezo-resistive strain gauges in a half-bridge configuration. This method is illustrated in U.S. Pat. No. 5,444,244, by Michael D. Kirk et al., issued Aug. 22, 1995). (3) The least common method is the capacitance method (using a standard capacitance bridge). Both normal and lateral (friction) forces acting on the tip can in principle be measured by any of these methods. In cases where friction forces are measured, the AFM is often referred to as a Friction Force Microscope (FFM).
When making xe2x80x9cforce measurementsxe2x80x9d at different locations of a surface (i.e., on scanning) with an AFM or FFM tip, one is also recording topographical images of the surface, i.e., using the tip as a microscope (hence the origin of the name Atomic Force Microscope and Friction Force Microscope).
The limitations of the prior art may be best understood by first considering the equations that describe the response of a simple cantilever spring to different kinds of forces, both normal and lateral, that give rise to different kinds of spring deflections (such as bending, twisting, and buckling). Considering the equations that pertain to optical and resistive detectors, it can be seen that these two detection systems have inherently different sensitivities to the spring deformation. This analysis also explains why similarly shaped cantilever springs may exhibit high sensitivity to lateral forces when measured with the optical technique, but low sensitivity when measured with a resistive bridge. The criteria needed for a cantilever spring or spring system to have high resolution for both normal and lateral forces can then be established. The present invention illustrates these principles with a new cantilever design that constitutes the preferred embodiment of the invention.
FIG. 1A shows a xe2x80x9csimple cantileverxe2x80x9d spring of length L, width b, and thickness t. The spring is clamped at one end, with a rigid tip of length h at the free end. This is the basic design of a typical AFM or FFM cantilever, although other versions, for example, triangular or double (side-by-side) cantilevers (FIG. 1B and FIG. 1C), are more commonly used. These design modifications, however, do not change the basic analysis presented here regarding the optimization of cantilever design to measure normal and lateral forces independently and at high sensitivity, which is the object of this invention.
When a normal force Fz or a lateral force Fx or Fy acts normally or horizontally on the tip at point P in FIG. 1A, the angular deflections xcex94xcex8 at Q may be determined according to the following equations (where xcex8ij refers to deflections of the cantilever about the j-axis due to a force applied along the i-direction, and where E is the Elastic Modulus of the cantilever material):
xcex94xcex8zx=L2Fz/2Ebh3 in bending mode due to a vertical force Fzxe2x80x83xe2x80x83(1)
xcex94xcex8xy=LhFx/Ebh3 in twisting mode due to a lateral force Fxxe2x80x83xe2x80x83(2)
xcex94xcex8yx=3LhFy/Ebh3 in buckling mode due to a lateral force Fyxe2x80x83xe2x80x83(3)
Since in OPTICAL mode the light beam bounces off the surface at Q, the above angles give the angle by which the light is deflected (the total change of angle on reflection is actually 2xcex94xcex8), which in turn is proportional to the sensitivity by which the forces can be measured using a quadrant photodetector. Since in many applications we have Fx≈Fy≈Fz (i.e., for friction coefficients close to 1), we see that high sensitivity to lateral forces (in twisting and buckling modes) compared to normal forces (in bending mode) can be achieved only for high values of h/L (h/L≈1).
In the case of cantilevers in RESISTIVE mode, the angular deflections at the tip P or surface Q given by Eqs. 1-3 are not what is measured by the resistive elements on the cantilever. (If resistive elements were to be placed at Q, they would hardly measure anything when the spring bends because the surface at Q is essentially rigid to local bending and remains flat when the other parts of the cantilever bend). Instead, the resistive elements must be placed along the compliant length of the cantilever, as in the Kirk et al. reference, U.S. Pat. No. 5,444,244. The cantilever bends into arcs of circles whose relevant angles of curvature are given by the following equations (cf. FIG. 1A):
xcex94xcex8zxxe2x88x9dL2Fz/2Ebh3 in bending mode due to a vertical force Fzxe2x80x83xe2x80x83(4)
xcex94xcex8xxxe2x88x9d2hFxEh3 in twist-bending mode due to a lateral force Fxxe2x80x83xe2x80x83(5)
xcex94xcex8yxxe2x88x9d3LhFy/Ebh3 in buckling mode due to a lateral force Fyxe2x80x83xe2x80x83(6)
These angles are proportional to the sensitivity by which the forces can be measured using a resistance detector such as a Wheatstone bridge, and it should be noted that whereas Equations (4) and (6) are proportional to Equations (1) and (3), the expression for the twist-bending deflection, Equation (5), is different from Equation (2), the equation for pure twist. Thus, when measured with a resistive cantilever, high sensitivity to lateral forces in twist-bending mode compared to normal forces (in bending mode) can be achieved only for high values of 2hb/L2. This compares with the factor of h/L for the optical detection method.
Thus, if the tip length h is much smaller than L, which is the norm in all current cantilevers used in AFM""s and FFM""s, the sensitivity to measuring friction forces in the twist mode using the OPTICAL method will be less than the sensitivity to measuring normal forces (by a factor that is proportional to h/L). But using the RESISTIVE method, the loss in sensitivity is usually even worse, being now proportional to 2hb/L2 (unless b greater than L/2). The inventor believes that this is the reason for the very low sensitivity of current resistive cantilevers when measuring lateral forces, so much so that resistive cantilevers are not in common use for such purposes.
It may appear that this problem may be solved by increasing the length of the tip h and cantilever width b, i.e., making h≈b≈L. However, increasing h introduces a new problem. As shown in FIG. 2A, a longer tip at the end of a cantilever spring will move significantly out of the vertical when a normal (vertical) force Fz or displacement xcex94z is applied to the tip. If the tip can slip (slide) freely along the surface, it will move along the arc of a circle of radius R=(h2+L2/4)xc2xd centered about the point O, which takes it further away from the vertical line for larger values of h. Only for small values of h (when h/L less than  less than 1) will this effect be unimportant, i.e., will the tip move perfectly vertically in the z-direction and so remain at the same x-y coordinates on the surface, at least for small deflections. However, for h≈L, the lateral displacement and tip bending that inevitably accompany the vertical displacement makes this solution impractical for unambiguous normal force measurements, at least when using the simple cantilever design shown in FIGS. 1A, 1B, and 1C.
In addition, depending on how xe2x80x98stuckxe2x80x99 the tip is to the surface, i.e., depending on the friction force between them, the tip deflection may be as shown in FIG. 2A or FIG. 2B or somewhere in between these two limits. Furthermore, the deflections may be different for upward and downward deflections (positive or negative z displacements) and therefore not even symmetrical about z=0.
A further limitation of the simple cantilever constructions illustrated in FIGS. 1A-C is that it is not obvious that purely vertical or lateral forces provide pure and independent bending along the different cantilever axes even when the tip length h is small. This is because of the complicated triangular or rectangular geometries of these commonly-used cantilevers.
Thus, with any geometry based in the simple cantilever construction shown in FIGS. 1A-C and FIGS. 2A-B, high resolution normal and lateral force measurements will always be coupled, and it will be impossible to measure one independently without also measuring some contribution from the other. This also implies that surface images will be likewise distorted and difficult to interpret.
It is therefore an objective of the present invention to provide a new cantilever design, based on the above equations, that will allow for independent measurements of tip-surface forces in both normal and lateral directions with similar high sensitivity for both measuring modes. The introduction of such a cantilever design would significantly enhance the ease of use and accessibility of AFM""s and FFM""s, and SFA""s having an XYZ scanner (such as the one described in the inventor""s U.S. Pat. No. 6,194,813, Issued Feb. 27, 2001) to carry out delicate friction force measurements.
In summary, the present invention is a dual- and triple-mode cantilever suitable for simultaneously measuring both normal (adhesion) and lateral (friction) forces independently in three orthogonal directions. The cantilever design allows the measurements to be performed at high sensitivity. The cantilever is useful in Scanning Probe Microscopes (SPM""s) and other force-measuring devices, such as the Atomic Force Microscope (AFM), the Friction Force Microscope (FFM) and in probe attachments for the Surface Forces Apparatus (SFA) where both normal and lateral forces acting on a tip need to be accurately and unambiguously measured. The resistive cantilever structure may also be used for optical detection of tip deflections.
An advantage of the present invention is that it allows three orthogonal forces to be measured independently at the same time. Prior art devices only allow the simultaneous measurement of two orthogonal forces.
Another advantage of the present invention is that it provides a system that has a higher general sensitivity to measuring forces than the prior art devices by making use of a full-bridge configuration, rather than a half-bridge construction. This doubles the intrinsic electric sensitivity of the bridge. The present invention also effectively eliminates differential thermal drifts because all four resistance elements are located in close proximity. With the half-bridge configuration, two of the resistors are located far from the cantilever resistors, which makes it more difficult to balance the bridge.
A still further advantage of the present invention is that it enables a symmetrical design of the cantilever system that ensures that all force-detecting modes (bending, twisting, buckling, and twist-bending) will respond independently to tip forces acting along the x, y, and z directions.
Still another advantage of the present invention is that the new cantilever may also be used with the optical method of detecting displacement, wherein a light beam reflects off the cantilever surface at some suitable point (not necessarily the center).